There's so many directions one can take with good pickled beats. Personally, I lurve me some creaminess with traditional flavored pickled beats, but the added spices put a bit of curve into the pitch, so it's one of those things that I think I'd have to taste to really be able to pair up nicely. Care to send me a jar for the sake of better suggestions?
If the additional spices add an undertone bite to the beets, you might be safe taking a more traditional East European route with them. A citrus zest creme fraiche dollop, quenelle, or sauce for a salmon dish would be wonderful, maybe with some asparagus.
You could go the salad route and still get some creaminess in there with a thick buttermilk or bleu cheese dressing with a peppery and/or bitter lettuce base.
If they have a strong spiced flavor, you might consider pairing it with orange and scallops, light fish, chicken, or pork (really, most neutral flavored meats), or keeping them as the main flavor component of a salad with some apple or pear and pecan accents, a light oil drizzle with generous seasoning.
Typically chevre goes well with heavy spicing, sweet tones, and deep savory bits, so that might be worth a consideration.